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DOI: 10.4230/LIPIcs.MFCS.2022.78

An Exact Algorithm for Knot-Free Vertex Deletion

Authors: M. S. Ramanujan, Abhishek Sahu, Saket Saurabh, and Shaily Verma

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract

The study of the Knot-Free Vertex Deletion problem emerges from its application in the resolution of deadlocks called knots, detected in a classical distributed computation model, that is, the OR-model. A strongly connected subgraph Q of a digraph D with at least two vertices is said to be a knot if there is no arc (u,v) of D with u ∈ V(Q) and v ∉ V(Q) (no-out neighbors of the vertices in Q). Given a directed graph D, the Knot-Free Vertex Deletion (KFVD) problem asks to compute a minimum-size subset S ⊂ V(D) such that D[V⧵S] contains no knots. There is no exact algorithm known for the KFVD problem in the literature that is faster than the trivial O^⋆(2ⁿ) brute-force algorithm. In this paper, we obtain the first non-trivial upper bound for KFVD by designing an exact algorithm running in time 𝒪^⋆(1.576ⁿ), where n is the size of the vertex set in D.

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M. S. Ramanujan, Abhishek Sahu, Saket Saurabh, and Shaily Verma. An Exact Algorithm for Knot-Free Vertex Deletion. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 78:1-78:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ramanujan_et_al:LIPIcs.MFCS.2022.78,
  author =	{Ramanujan, M. S. and Sahu, Abhishek and Saurabh, Saket and Verma, Shaily},
  title =	{{An Exact Algorithm for Knot-Free Vertex Deletion}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{78:1--78:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.78},
  URN =		{urn:nbn:de:0030-drops-168769},
  doi =		{10.4230/LIPIcs.MFCS.2022.78},
  annote =	{Keywords: exact algorithm, knot-free graphs, branching algorithm}
}

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DOI: 10.4230/LIPIcs.IPEC.2021.23

A Polynomial Kernel for Bipartite Permutation Vertex Deletion

Authors: Lawqueen Kanesh, Jayakrishnan Madathil, Abhishek Sahu, Saket Saurabh, and Shaily Verma

Published in: LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

Abstract

In a permutation graph, vertices represent the elements of a permutation, and edges represent pairs of elements that are reversed by the permutation. In the Permutation Vertex Deletion problem, given an undirected graph G and an integer k, the objective is to test whether there exists a vertex subset S ⊆ V(G) such that |S| ≤ k and G-S is a permutation graph. The parameterized complexity of Permutation Vertex Deletion is a well-known open problem. Bożyk et al. [IPEC 2020] initiated a study towards this problem by requiring that G-S be a bipartite permutation graph (a permutation graph that is bipartite). They called this the Bipartite Permutation Vertex Deletion (BPVD) problem. They showed that the problem admits a factor 9-approximation algorithm as well as a fixed parameter tractable (FPT) algorithm running in time 𝒪(9^k |V(G)|⁹). And they posed the question {whether BPVD admits a polynomial kernel.} We resolve this question in the affirmative by designing a polynomial kernel for BPVD. In particular, we obtain the following: Given an instance (G,k) of BPVD, in polynomial time we obtain an equivalent instance (G',k') of BPVD such that k' ≤ k, and |V(G')|+|E(G')| ≤ k^{𝒪(1)}.

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Lawqueen Kanesh, Jayakrishnan Madathil, Abhishek Sahu, Saket Saurabh, and Shaily Verma. A Polynomial Kernel for Bipartite Permutation Vertex Deletion. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kanesh_et_al:LIPIcs.IPEC.2021.23,
  author =	{Kanesh, Lawqueen and Madathil, Jayakrishnan and Sahu, Abhishek and Saurabh, Saket and Verma, Shaily},
  title =	{{A Polynomial Kernel for Bipartite Permutation Vertex Deletion}},
  booktitle =	{16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-216-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{214},
  editor =	{Golovach, Petr A. and Zehavi, Meirav},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.23},
  URN =		{urn:nbn:de:0030-drops-154065},
  doi =		{10.4230/LIPIcs.IPEC.2021.23},
  annote =	{Keywords: kernelization, bipartite permutation graph, bicliques}
}

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Documents authored by Verma, Shaily

An Exact Algorithm for Knot-Free Vertex Deletion

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A Polynomial Kernel for Bipartite Permutation Vertex Deletion

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